PHU 205 Mechanics for Life Sciences

Chapter I:

Introduction Prof. Liliana Braescu & Prof. Nouredine Zettili

Required text College Physics Raymond A. Serway and Chris Vuille 9th Edition, 2012 BROOKS/COLE CENGAGE Learning ISBN 10: 1-111-42745-3 ISBN 13: 978-1-111-42745-0

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Course Outline Chapter 1

Introduction 1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.1 Standards of Length, Mass and Time Theory and Experiments u The goal of physics is to develop theories based on experiments u A physical theory, usually expressed mathematically, describes how a given system works u The theory makes predictions about how a system should work u Experiments check the theories’ predictions u Every theory is a work in progress

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Fundamental Quantities and Their Dimension u To describe motion, physicists use three fundamental quantities: length, mass, and time

✔ length (L) 10-20 m ≤ L ≤ 15 billion light years ✔ mass (M) 10-31 kg (me) ≤ M ≤ 1052 kg ✔ time (T) 0 seconds ≤ T ≤ 15 billion years u Other physical quantities can be constructed from these three

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Systems of Measurement u In Physics we employ two main unit systems:

SI (le Systeme International) or MKS system international unit system: 1960 Paris Conference standard units of measurement include

✔meter (m) for length ✔kilogram (kg) for mass ✔second (s) for time

u Gaussian or CGS system -unit system commonly used in electromagnetism -standard units of measurement include

✔centimeter (cm) for length ✔gram (g) for mass ✔second (s) for time

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Units in Various Systems

System Length Mass Time

SI meter kilogram second CGS centimeter gram second US Customary foot slug second

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.2 The Building Blocks of Matter u Matter exists on several different size scales:

✔ Virus ≈ 10-7 m ✔ Molecule ≈ 10-9 m ✔ Atom ≈ 10-10 m ✔ Nucleus ≈ 10-14 m ✔ Protons & Neutrons ≈ 10-15 m ✔ Quarks ≈ 10-18 m ✔ Sub-quark ≈ ???

! ! ! strong evidences indicated that protons, neutrons and a zoo of other exotic particles are composed of six particles called quarks

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Remarks: u Notice that the average size of the atom is about 1 Angstrom (Å) or 10-10 meters. u Within the atom’s center, the nucleus is only about 10-14 meters in size. u This means that the electron cloud surrounding the nucleus occupies an enormous amount of space in relation to the atom as a whole. u Indeed, all of the electronic activity of the universe occurs in this vast emptiness. u If Earth were stripped of all of its electrons, its radius would reduce to merely half a mile! u This would put Earth into a class of stellar bodies known as neutron stars. u If this were reality, human life as we know it could not exist.

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.3 Dimensional Analysis u A dimension of a quantity is the physical nature of that quantity. u The fundamental dimensions are expressed by

“L” for length “M” for mass “T” for time

u Every quantity can be broken down into these three fundamentals. u Only quantities of the same dimension can be added/subtracted. u Any two quantities can be multiplied/divided regardless of their physical dimensions.

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1.3 Dimensional Analysis u When expressing a value in dimensional form, place it in brackets. Examples:

2[ ]area L L L= × = [ ] LvelocityT

=

3[ ]volume L L L L= × × =2

[ ] LaccelerationT

=

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Remarks: u Physical dimensional analysis can be used to check the validity of an equation. u In order for an equation to be physically correct, it must have the same dimensions on both sides. Example:

Is this equation physically possible? NO!

VxT

=

[ ][ ][ ]VxT

=LTLT

= 2

LLT

≠

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Problems u Problem 1.1

The period of a simple pendulum, defined as the time for one complete oscillation, is measured in time units and is given by

where ℓ is the length of the pendulum and g is the acceleration due to gravity, in units of length divided by time squared. Quest ion: Show that th is equat ion is dimensionally consistent.

2Tg

π= l

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Problems u Problem 1.1 Solution

Substituting in the fundamental dimensions, Thus, the dimensions are consistent.

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[ ][ ][ ] /l LT T Tg L T

= = = =

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.4 Uncertainty in Measurement and Significant Figure

u There is uncertainty in every measurement, this uncertainty carries over through the calculations

Need a technique to account for this uncertainty

u We will use rules for significant figures to approximate the uncertainty in results of calculations

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Significant Figures u A significant figure is a reliably known digit u All non-zero digits are significant u Zeros are not significant when they only locate the decimal point

Using scientific notion to indicate the number of significant figures removes ambiguity when the possibility of misinterpretation is present

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Operations with Significant Figures u When multiplying or dividing two or more quantities, the number of significant figures in the final result is the same as the number of significant figures in the least accurate of the factors being combined

Least accurate means having the lowest number of significant figures

u When adding or subtracting, round the result to the smallest number of decimal places of any term in the sum (or difference)

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Rounding u Calculators will generally report many more digits than are significant

Be sure to properly round your results u Slight discrepancies may be introduced by both the rounding process and the algebraic order in which the steps are carried out

Minor discrepancies are to be expected and are not a problem in the problem-solving process

u In experimental work, more rigorous methods would be needed

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.5 Conversion of Units

u Within the metric unit system (SI and CGS), all quantities are based on powers of ten. u Convers ion requires simply moving the decimal. u Table 1.4 lists several prefixes for the powers of ten used in the metric system.

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1.5 Conversion of Units u Converting between SI and English units requires the use of conversion factors. Examples:

1 mile = 1609 meters 1 foot = 0.3048 meters 1 inch = 2.54 centimeters

u Many other conversion factors can be found in the front cover of your textbook. Example:

Express 60 mph in SI units.

60 1609 1 26.82 /1 1 3600miles m hour m shour mile s

⎛ ⎞⎛ ⎞ =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

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Problems (for more examples see the Book - Chapter 1)

u Problem 1.2

A house is 50.0 ft long and 26.0 ft wide, and has 8.0-ft-high ceilings. Question: What is the volume of the interior of the house in cubic meters and in cubic centimeters?

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Problems u Problem 1.2 Solution

u Convert each length to SI units: 50.0 ft = 15.2 m = 1520 cm 26.0 ft = 7.95 m = 7950 cm 8.0 ft = 2.44 m = 244 cm

u Vol. of house = L x W x H = (15.2 m)(7.95 m)(2.44 m) = 295 m3

u Vol. = (1520 cm)(7950 cm)(244 cm) = 2.95 x 108 cm3

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.6 Estimates and Order-of-Magnitude Calculations

Estimates u Can yield useful approximate answers

An exact answer may be difficult or impossible Mathematical reasons Limited information available

u Can serve as a partial check for exact calculations

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1.6 Estimates and Order-of-Magnitude Calculations

Order of Magnitude u Approximation based on a number of assumptions

May need to modify assumptions if more precise results are needed

u Order of magnitude is the power of 10 that applies

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.7 Coordinates Systems

u Any coordinate system is defined by the following:

–  an origin, O (a fixed reference point)

–  a set of scaled, labeled axes –  a method of defining any point in

space in relation to the origin u In the Cartesian coordinates system, any point p is defined by the ordered pair (x,y). Example: In this diagram, point Q is defined by an x-coordinate (-3) and a y-coordinate (4).

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1.7 Coordinates Systems

u In the Polar Coordinates system, any point p is defined by the ordered pair (r,θ). •  r is the distance from the

origin O •  θ is the angle between the

reference line and r. u Car tes ian and polar coordinates are re lated according to these equations:

cosx r θ= siny r θ=

2 2r x y= + arctan yx

θ =

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Chapter 1: Introduction

1.1 Standards of Length, Mass and Time 1.2 The Building Blocks of Matter 1.3 Dimensional Analysis 1.4 Uncertainty in Measurement and Significant Figures 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude-Calculations 1.7 Coordinate Systems 1.8 Trigonometry

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1.8 Trigonometry

u Tr i g o n o m e t r y i s t h e mathematical approach to studying angles. u A c c o r d i n g t o t h e Pythagorean Theorem, the sum of the squares of the two sides of any right triangle will always equal the square of the hypotenuse:

2 2 2a b c+ =

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Trigonometric Functions

The acute angles of a right triangle are complementary.

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The quadrants

The unit circle approach

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Extending from Quadrant I to the other quadrants

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Fundamental trigonometry identities

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Fundamental trigonometry identities

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Graphs of basic trigonometric functions